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A micromechanical model for predicting the fracturetoughness of functionally graded foams

Seon-Jae Lee 1, Junqiang Wang 1, Bhavani V. Sankar *

Department of Mechanical and Aerospace Engineering, P.O. Box 116250, University of Florida, Gainesville, FL 32611-6250, USA

Received 11 June 2006; received in revised form 3 November 2006Available online 10 November 2006

Abstract

In this paper, finite element method based micromechanical analysis is used to understand the fracture behavior of func-tionally graded foams. The finite element analysis uses a micro-mechanical model in conjunction with a macro-mechanicalmodel in order to relate the stress intensity factor to the stresses in the struts of the foam. The stress intensity factor at thecrack tip of the macro-mechanical model can be evaluated using either the J-contour integral or the stresses in the singu-larity-dominated zone. The fracture toughness is evaluated for various crack positions and length within the functionallygraded foam. Then the relationship between the fracture toughness of the graded foam and the local density at the cracktip is studied. Convergence tests for both macro-mechanical and micro-mechanical model analysis were conducted in orderto maintain adequate accuracy with reasonable computational time. Fracture toughness of homogenous foams and func-tionally graded foams for various cases are presented as a function of relative density. This study indicates that the fracturetoughness of functionally graded foams mainly depends on the relative density at the crack-tip.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Cellular materials; Fracture simulation; Fracture toughness; Functionally graded materials; Graded foams; Inhomogeneousfoams; Micromechanics

1. Introduction

The microstructure of cellular solids such as metallic and carbon foams can be tailored to obtain optimumperformance for use in multi-functional structures. Such multi-functional structures can be used in integralload-carrying thermal protection systems for hypersonic vehicles due to their low thermal conductivity,increased strength and stiffness. An excellent treatise on structure and properties of cellular solids can be foundin Gibson and Ashby (1998). They have presented approximate formulas for Mode I fracture toughness ofcellular solids as a function of relative density and the strength of the strut or cell wall material. Choi and San-kar (2003, 2005) found that the Mode I and II fracture toughness of cellular solids strongly depend on the

0020-7683/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijsolstr.2006.11.007

* Corresponding author. Tel.: +1 352 392 6749; fax: +1 352 392 7303.E-mail address: sankar@ufl.edu (B.V. Sankar).

1 Graduate student.

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relative-density; however the cell spacing and strut dimensions also have an effect on the fracture toughness.Although significant work has been done in modeling homogeneous cellular materials, models for functionallygraded foams or graded cellular materials, especially for strength and fracture toughness, are at their infancy,and it will be the main focus of this paper. Functionally graded foams can be made by controlling the size ofthe voids in the porous medium. The density can be varied either by changing the unit-cell dimensions or strutcross-section or both. Graded foams can also be manufactured by dispersing hollow micro-balloons of varyingsizes in a matrix medium (Madhusudhana et al., 2004). In this paper, both cases—varying wall thickness andvarying unit-cell dimensions—are considered independently using a finite element based micro-mechanicalanalysis. The finite element analysis uses a micro-mechanical model in conjunction with a macro-mechanical,or simply macro-model, in order to relate the stress intensity factor to the stresses in the struts of the foam.The stress intensity factor of the macro-model at the crack tip is evaluated using the J-contour integral. Thefracture toughness is evaluated for various crack positions and crack lengths within the functionally gradedfoam. Then the relationship between the fracture toughness of the functionally graded foam and the local den-sity at the crack tip is studied. In addition, convergence tests for both macro- and micro-models were conduct-ed. The results will help in understanding the fracture behavior of functionally graded foams.

2. Overview of the procedures

The method of determining the fracture toughness of the graded foam described in this paper is similar tothat discussed by Choi and Sankar (2003, 2005) for homogeneous foams. In the present case, the cellular mate-rial is inhomogeneous in the macro-scale. That is, the microstructure is graded and the foam is treated as afunctionally graded material in macro-scale. The foam can be modeled either as an inhomogeneous continu-um, or as a frame consisting of beam elements to model the struts. The former model (continuum model) willbe referred to as the macro-model and the latter (frame model) as the micro-model. In the finite element anal-ysis solid element are used in the macro-model and beam elements in the micro-model.

We create the crack in the foam by removing a set of struts along the intended crack surface. We consider aportion of the foam surrounding the crack tip for the micro-model (see Fig. 1). The dimensions of the micro-model should be much larger than the cell size (strut spacing) so that it can be considered as a continuum.Then we apply displacements to the boundary nodes of the micro-model such that a known stress intensityfactor (SIF) exists at the crack tip. The maximum stresses in the struts in the vicinity of the crack tip are cal-culated from the FE micro-model. From the failure criterion for the strut material, one can calculate the max-imum stress intensity factor that will cause the failure of the crack tip struts, and thus causing crackpropagation in a macro-scale sense. The key thing in this approach is to be able to calculate the SIF for a given

h

h

c

Crack

x

y

Width

Macro-model

Micro-model

v0

A B

CD

a b

Fig. 1. (a) Macro-model consists of plane 8-node solid elements. The region in the middle with grids indicates the portion used in themicro-model. (b) The micro-model consists of frame elements to model the individual struts. The displacements from the macro-model areapplied as boundary conditions in the micro-model.

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boundary displacements or apply a set of boundary conditions that corresponds to a given stress intensity fac-tor in the macro-scale sense.

For this purpose we turn to the macro-model as shown in Fig. 1. In the macro-model a much larger size of thefoam is modeled using continuum elements, in the present case, plane solid elements. The micro-model is basi-cally embedded into the macro-model. The foam is subjected to an arbitrary boundary condition to produce agiven mode-mixity at the crack tip. In this paper, we consider only Mode I symmetric loading (mode-mixity = 0).The stress intensity factor at the crack tip is calculated either from the crack-tip stress field or the J-integral. Thedisplacements of points along the boundary of the micro-model are obtained from the FE analysis of the macro-model and applied to the boundary of the micro-model as explained in the preceding paragraph.

It should be noted that the aforementioned macro-model was not necessary in the case of homogeneousfoams (Choi and Sankar, 2003). Closed-form expressions for crack-tip displacement field are available fororthotropic materials (Sih and Liebowbitz, 1968), and hence the displacement boundary conditions for themicro-model for a given SIF are easily obtained.

3. Estimation of continuum properties

The macro-model of the graded foam requires continuum properties at each point or at least for each ele-ment in the finite element model. In the following, we briefly describe the procedures to calculate the contin-uum properties of a homogeneous cellular medium and then extend the idea to graded foams.

3.1. Homogeneous foam

Most of the open-cell foams can be considered as orthotropic materials. Choi and Sankar (2005) derived theelastic constants of homogeneous foams in terms of the strut material properties and unit-cell dimensions. Intheir model they assumed that the strut has a square cross-section and the unit-cell is a cube. In the presentapproach we would like to consider a general case wherein the unit-cell is a rectangular parallelepiped ofdimensions c1 · c2 · c3. The derivation of formulas for the density and Young’s modulus are straightforwardand they are as follows:

q�

qs

¼ðc1 þ c2 þ c3Þh2 � 2h3

c1c2c3

ð1Þ

E�1 ¼h2

c2c3

� �Es; E�2 ¼

h2

c1c3

� �Es; E�3 ¼

h2

c1c2

� �Es ð2Þ

where a superscript * denotes the foam properties and a subscript s denotes the solid properties or the strutproperties.

The derivation of shear modulus is slightly involved and it is described below. We show the derivation ofthe shear modulus G�12 from the unit-cell dimensions, strut cross-sectional dimensions and the strut Young’s

c1/2

c2/2

d1

Curvature=0

F1

F2

d2c2/2

c1/2

Fig. 2. Flexural deformation of struts under shear stresses.

S.-J. Lee et al. / International Journal of Solids and Structures 44 (2007) 4053–4067 4055

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modulus. When a shear stress is applied, struts are deformed as shown in Fig. 2. Bending moment becomeszero at the middle of struts because the curvatures are zero due to symmetry. The struts are assumed as a beamfixed at the end with a concentrated force at the middle at a distance c1

2and c2

2, respectively from the fixed end as

shown in Fig. 2. The maximum displacement can be written as,

d1 ¼PL3

3EI¼

F 2c1

2

� �3

3EsIand d2 ¼

PL3

3EI¼

F 1c2

2

� �3

3EsIð3Þ

where I ¼ bh3

12¼ h4

12

The applied shear stress can be written as s�12 ¼ F 1

c1c3¼ F 2

c2c3. Using the relations, F 1

c1¼ F 2

c2, the maximum displace-

ments (Eq. (3)) can be rewritten as

d1 ¼c3

1F 2

24EsIand d2 ¼

c1c22F 2

24EsIð4Þ

The shear strain c12 can be derived as

c12 ¼2d2

c2

þ 2d1

c1

¼ 2c1d2 þ 2c2d1

c1c2

ð5Þ

Using Eq. (4), the shear strain can be written as,

c12 ¼ðc1c3 þ c2

1ÞF 3

12EsIð6Þ

The shear modulus G�12 can be derived as

G�12 ¼s�12

c12

¼F 2

c2c3

c1c2þc21ð ÞF 3

12EsI

¼ 12EsIc2c3 c1c2 þ c2

1ð Þ ð7Þ

Substituting for I we obtain

G�12 ¼h4

c1c2c3ðc1 þ c2Þ

� �Es ð8Þ

The shear modulus in the other two planes can be obtained by cyclic permutation as

G�23 ¼h4

c1c2c3ðc2 þ c3Þ

� �Es

G�31 ¼h4

c1c2c3ðc1 þ c3Þ

� �Es ð9Þ

3.2. Graded foam

The properties of a graded foam can be represented by a function of the coordinate variables x, y and z. Theactual functional form depends on the application and also the type of information sought from the homog-enized model of the foam. In this study, we will assume that the functions will be such that the properties cal-culated at the center of a cell will correspond to the properties of the homogeneous foam with that cell as itsunit cell. Thus the function is actually defined only at the centers of the cells of the graded foam. Then, we willfit an equation through these points to obtain the continuous variation of properties required in the contin-uum model. This approach will be verified by solving some problems wherein the graded foam is subjected tosome simple loading conditions and comparing the results from the macro- and micro-models.

In this study, the strut properties are taken as that of carbon foam studied earlier (Choi and Sankar, 2003),and they are listed in Table 1.

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The relative density of functionally graded foams (FGF) depends on both the dimensions of the unit-celland the strut thickness. Therefore, three different cases can be considered. The first case is the one wherethe dimensions of the unit-cell remain constant while the strut thickness varies along the x-axis. In the secondcase the strut thickness is kept constant with varying cell length. The last case is varying both of them. In thispaper, the first two cases are studied independently. Furthermore, the material properties of functionally grad-ed foam can be either increasing or decreasing along the x-axis. Therefore, the fracture properties of bothincreasing and decreasing cases are studied and compared to the homogenous case.

For the case where the strut dimensions vary, the thickness of the square strut is assumed to vary as

hðxÞ ¼ h0 þ ax ð10Þwhere a is a parameter that determines the degree of gradation of the properties. Then the properties such asdensity and elastic constants of the graded foam can be assumed to vary as given by the equations for homo-geneous foams, but changing the constant h by the function h(x). For example, the density variation of thefoam follows from Eq. (1) and can be written as

q�

qs

¼ 3hðxÞ

c

� �2

� 2hðxÞ

c

� �3

ð11Þ

where the unit-cell is assumed as a cube of dimension c. Similar equations can be derived for Young’s modulusand shear modulus as

E� ¼ hðxÞc

� �2

Es

G� ¼ 1

2

hðxÞc

� �4

Es ð12Þ

Similarly we can consider the case where h, c2 and c3 are constants, but c1 varies as

ciþ11 ¼ ci

1 þ b ð13Þwhere i denotes the cell number and b is the increment in the cell length in the x direction. Again the propertiesof the foam will be calculated at the center of each cell using the equations for homogeneous foams as given inEqs. (2)–(9) (see Fig. 3).

In order to verify the validity of the functional form of elastic constants that are used in the present study, asimple mechanics problem was solved using both macro- and micro-models. A rectangular solid made of theinhomogeneous foam was considered as shown in Fig. 1 (a). A uniform extension (70 lm) was applied alongthe upper edge of macro-model, which consists of two dimensional plane stress elements (eight nodes bi-qua-dratic, reduced integration element). The elastic constants of the inhomogeneous material varied as given byEq. (12). In the FE model the elastic constants within each element were considered constant. The boundaryconditions are depicted in Fig. 4. The right lower corner was fixed to prevent the rigid body motion. Theresulting displacements along the boundary of the micro-model, embedded in the macro-model, were appliedon the boundary of the micro-model by using the three-point interpolation. For the micromechanical model,each strut was modeled as an Euler–Bernoulli beam with two nodes and three integration point element. Inorder to verify the validity of properties used in the macro-model we compare the stresses in both models.In the case of macro-model the stresses are obtained as the FE output. The outputs in micro-model are theaxial and transverse forces in the beam element. We convert these forces into equivalent stresses by dividingthe strut cross-sectional area c1 · c3.

Table 1Properties of strut material

Density, qs 1750 kg/m3

Elastic modulus, Es 207 GPaPoisson’s ratio, ms 0.17Ultimate tensile strength, ru 3600 MPa

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x (m)

0.00 0.02 0.04 0.06 0.08 0.10

Elastic Modulus (Pa)

0.00

2.00e+9

4.00e+9

6.00e+9

8.00e+9

1.00e+10Relative density

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Elastic ModulusRelative density

(Kg/m3)

Fig. 3. Example of variation of elastic modulus and relative density for constant cell length c and a = �200 · 10�6.

Uniform Extension

ux=0, uy=0 ux 0, ≠ uy=0

A B

CD

Fig. 4. Boundary conditions for edge-cracked plate under uniform extension.

x (m )0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014

Stresses (MPa)

0

2

4

6

8

10

12

14

sub-macro-modelmicro-model

Approx. 5% difference

Fig. 5. Comparison of stresses (r22) obtained using the macro- and micro-models in graded foam with constant cell size but varying strutcross-section.

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Fig. 4 depicts the macro-, sub-macro- and micro-models. We consider both constant cell length and con-stant strut size cases. In the constant cell length case the cell length is assumed as 200lm. The macro-modelconsists of 100 · 50 plane solid elements. The strut cross-section is assumed to vary as a function of x accord-ing to the equation h(x) = h0 + ax, where h0 = 40lm and a = �200 · 10�6. The region corresponding to themicro-model in the macro-model consists of 15 · 5 elements. The micro-model uses 75 · 25 beam elements.The results for the stress component r22 from the macro- and micro-models are compared in Fig. 5. In thesecond case, the strut is assumed to have a square cross-section (h = 20 lm) and the cell length c1 was variedalong the x direction with c0

1 ¼ 200 lm and b = �0.15 lm. The dimensions of the cell in the 2 and 3 directions,c2 andc3, are kept constant (100 lm). The stress component r22 along the upper boundary of the micro-modelare compared in Fig. 6. The maximum difference in stresses between the macro- and micro-models is about5%.

4. Fracture analysis

4.1. Macro-mechanical model

In this section, the finite element analysis (FEA) of the functionally graded foam containing a crack is per-formed. The purpose of the analysis is to determine the displacements along the boundaries corresponding tothe micro-model for a given stress intensity factor. The macro-model for the foam is modeled using plane solidelements. An arbitrary boundary condition is applied to the macro-model and the corresponding stress inten-sity factor and displacements are found from the FE results. Due to symmetry about the crack plane only tophalf of the foam is modeled. A constant displacement in the vertical direction is applied along the top hori-zontal boundary as shown in Fig. 4. The J-Integral is calculated along several contours surrounding the cracktip. The displacements at the nodes on the boundary corresponding to the micro-model are obtained from themacro-model FEA results. The upper edge is loaded by a uniform displacement loading in the y-direction, andthe lower edge has a zero displacement boundary condition in the y-direction to account for symmetry. Thematerial is functionally graded by either changing the thickness of struts or changing the dimensions of theunit-cell described as before. Relative density, elastic modulus and shear modulus vary along the x-axis cor-responding to the equations derived in the previous section. When, functionally graded foam is modeled as ahomogeneous solid (macro-model), a material property discretization is introduced. Material properties arediscretized by assigning each region along the x-axis the value of those at the centroid of the region. For exam-

x (m)

0.000 0.005 0.010 0.015 0.020 0.025 0.030

Stresses (MPa)

0

2

4

6

8

sub-macro-modelmicro-model

Fig. 6. Comparison of stresses (r22) obtained using the macro- and micro-models in graded foam with constant strut size but varying celldimensions.

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ple, Fig. 7 shows the discrete elastic modulus for the 10-region model. However, the Poisson’s ratio is keptconstant because the effect of a variation of Poisson’s ratio is negligible (Delale and Erdorgan, 1983).

A two dimensional continuum element with 8 nodes and reduced integration points is used for the planestress problem. Generally, J-integral is not path independent for inhomogeneous material. Therefore, J-inte-gral is expected to vary with contour numbers as shown in Fig. 8. The contour numbers represent incremen-tally larger contours around the crack tip. The mesh refinement governs the size and increments of contours.However, the value of J-integral for a contour very close to the crack-tip is related to the local stress intensityfactor as in the case of a homogeneous material (Anals et al., 2000). Thus, energy release rate, G is identical tothe value of J-integral as the path of contour approaches to zero (Gu et al., 1999). Energy release rate, G canbe found by using 4th order polynomial regression of the variation of J-integrals as shown in Fig. 8. For theconvergence test, the model is discretized into uniform meshes of 10 · 5 elements (10 regions), 20 · 10 (20regions), 50 · 25 elements (50 regions), 100 · 50 (100 regions), 200 · 100 (200 regions) and 400 · 200 (400regions). As the number of elements and regions increases, the energy release rate at the crack tip convergesas shown in Fig. 9. For 100 · 50 elements model, the variation of J-integral is less than 0.01% compared to the400 · 200 element model. Therefore, 100 · 50 elements model is used for further analysis in order to maintainadequate accuracy with reasonable computational time.

The stress intensity factor KI of a functionally graded foam (two dimensional orthotropic) can be foundfrom G using the relation (Sih and Liebowbitz, 1968).

G ¼ K2I

a11a22

2

� �12 a22

a11

� �12

þ 2a12 þ a66

2a11

" #12

ð14Þ

where; a11 ¼1

E1

; a22 ¼1

E2

; a33 ¼1

E3

a12 ¼ a23 ¼ a31 ¼ 0

a44 ¼1

G23

; a55 ¼1

G13

; a66 ¼1

G12

The stress intensity factor at the crack tip can also be obtained directly from the crack-tip stresses as

KI ¼ Limr!0

ryyðr; 0Þffiffiffiffiffiffiffi2prp

ð15Þ

x (m)

0.00 0.02 0.04 0.06 0.08 0.10

Elastic Modulus (Pa)

0.00

2.00e+9

4.00e+9

6.00e+9

8.00e+9

1.00e+10

Actual elastic modulusDiscrete elastic modulus

Fig. 7. Discrete elastic modulus for macro-model with 10 regions.

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Fig. 10 shows the one of the example plot of ryy

ffiffiffiffiffiffiffi2prp

versus distance from the crack tip. A 4th order polyno-mial regression is also shown in Fig. 10. The y-intercept of the curve yields the value of KI.

The stress intensity factor from both methods, J-Integral and stress-matching, were compared for variouscases in Table 2. The maximum difference between the two methods is less than 6% for range of crack lengthsstudied here.

4.2. Micro-mechanical model

A portion of macro-mechanical model is taken and used for micro-model as shown in Fig. 4. As the100 · 50 elements (100 regions) for macro-model and constant cell length (100 lm) for micro-model are used,one macro-model element can be replaced by 100 elements micro-model (10 beam elements in x and y direc-tions). The displacements along the boundaries of micro-model are determined by using three points interpo-lation. The corresponding three points can be obtained in previous macro-model analysis. In micro-model,two-node beam elements are used to represent the foam ligaments/struts. The stresses at the crack tip canbe calculated from the results for force and moment resultants obtained from the micro-model as

y = 1.21193E-05x4 - 1.27727E-03x3 - 6.32919E-02x2 - 2.05304E+00x + 7.61841E+02

R2 = 9.99992E-01

0

100

200

300

400

500

600

700

800

1 6 11 16 21 26 31 36 41 46 51

Contour Number

J-integral

(J/m2)

Fig. 8. J-Integral for various contours in a macro-model containing 100 · 50 elements. Contour numbers increase away from the cracktip.

609

609.5

610

610.5

611

611.5

612

612.5

613

0 50 100 150 200 250 300 350 400

Number of regions

Energy release rate

(J/m2)

0

200

400

600

800

1000

1200

1400

Energy release rate

Time needed to complete job

Time (sec)

Fig. 9. Variation of energy release rate at the crack tip with various size macro-models.

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pyrtip ¼

M tiphtip

2

I tip

� F tip

Atip

ð16Þ

The fracture toughness of the foam is defined as the stress intensity factor that will cause the crack-tip struts tofail. We assume that the strut material is brittle and will fracture when the maximum principal stress exceedsthe tensile strength ru. Since we are dealing with linear elasticity the fracture toughness can be estimated fromthe following relation,

KI

KIc

¼ rtip

ruð17Þ

The convergence analysis is conducted to evaluate the variation of fracture toughness with various sizes ofmicro-model, 1 · 3 macro-model (300 elements in micro-model), 2 · 6 (1200), 5 · 15 (7500), 7 · 21 (14,700)and 10 · 30 (30,000). As model size increases, fracture toughness converges as shown in Fig. 11. For 7500beam element model, the error in fracture toughness is less than 0.3 % compared to 30,000-beam elementsmodel. Therefore, the 7500-model is chosen for further analysis to compromise between the accuracy andcomputational time. The Mode I fracture toughness with various relative densities is conducted in two differ-ent sets for the constant unit-cell case. The first set is controlling the crack length while the variation of mate-rial properties remains same. The other set is shown in Fig. 12. The crack length remains constant while thematerial properties are controlled to locate desired relative density at the crack tip. However, the dimensionsof models are fixed (0.1m by 0.5m for macro-model and 0.015m by 0.005m for micro-model). For the casewhere the unit-cell dimensions change, the number of elements both in macro-model (100 · 50 elements)and micro-model (150 · 50 elements) are fixed and the material properties at the crack tip is controlled byc0

1 and b. Therefore, the dimensions of models are not fixed.

0.0E+00

1.0E+05

2.0E+05

3.0E+05

4.0E+05

5.0E+05

6.0E+05

7.0E+05

8.0E+05

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

r (m)

0.5( )Pa m⋅

syy√2πr

Fig. 10. Stress intensity factor from the stresses normal to the crack plane.

Table 2Comparison between two methods

Crack length (m) 0.01 0.02 0.03 0.04 0.05

Relative density at the crack-tip 0.094582 0.085536 0.076874 0.068608 0.06075KI From J-Integral (Pa-m1/2) 1.22381E+06 1.06997E+06 9.28935E+05 8.48869E+05 7.26919E+05KI From crack-tip stresses (Pa-m1/2) 1.22620E+06 1.07152E+06 9.33605E+05 8.06059E+05 6.83636E+05% difference 0.195 0.145 0.503 5.043 5.954

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5. Results and discussion

The fracture toughness of graded foams are compared to that of homogeneous foams with density same asthe crack-tip density of graded foams. Both cases, increasing and decreasing densities, are considered. Theresults for c = 200 lm and for different strut thicknesses are given in Tables 3 and 4, and for c = 100 lm inFig. 13. As seen from these tables and figure, the results from the present analysis are very close to thoseof homogeneous foam. However we see an interesting trend in Fig. 13. In both deceasing and increasing den-sity cases, the fracture toughness deviates from that of uniform density foam for higher densities. When thedensity decreases along the crack path, the fracture toughness is slightly higher and vice versa.

The graded foams have constant unit-cell length (c = 200 lm) and the density is varied by changing the lig-ament cross-sectional dimensions. Results for the case of varying unit-cell dimensions are presented in Table 5

Fig. 12. Location of model specimens in the global panel. Each specimen is of the same size and contains a crack of given length, but thedensity at the crack tip varies from specimen to specimen.

0.00E+00

2.00E+05

4.00E+05

6.00E+05

8.00E+05

1.00E+06

1.20E+06

1.40E+06

1.60E+06

0 5000 10000 15000 20000 25000 30000

Number of elements

Fracture

toughness(Pa-m0.5)

0

50

100

150

200

Fracture toughness

time needed to compelete job

Time (Sec)

Fig. 11. Variation of fracture toughness with the size of micro-models.

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and also shown in Fig. 14. The results are also verifying the quite accurate results for the case where the dimen-sions of the unit-cell changed with constant cross-sectional area of the struts.

6. Concluding remarks

In this study, fracture toughness of functionally graded foams is predicted by employing both macro- andmicro- finite element models. The macro-model in which the foam is modeled as a continuum is used to esti-mate the stress intensity factor at the crack tip. Both stress matching method and J-Integral are employed tocompute the stress intensity factor for a given loading of the foam. The micro-model uses beam elements tomodel the struts and to determine the maximum stresses corresponding to the macro-stress intensity factor.From the failure criterion for the strut material the fracture toughness of the foam is estimated. It is foundthat the fracture toughness of functionally graded foam is approximately the same as that of homogeneousfoam of same microstructure with the same density at the crack tip.

There are not many experimental works available for fracture toughness of graded foams. Recently El-Hadek and Tippur (2003) have developed a novel method to fabricate graded foams by dispersing microbal-

Table 3Fracture toughness of graded and uniform foams

h0 (lm) Relative density at the crack-tip Fracture toughness (Pa-m1/2)

Decreasing density Increasing density Uniform density

26 0.028 4.52171E+05 4.56445E+05 4.51326E+0530 0.039744 6.56122E+05 6.57406E+05 6.56114E+0550 0.123904 2.24739E+06 2.25108E+06 2.24928E+0660 0.179334 3.39537E+06 3.39999E+06 3.39819E+0670 0.241664 4.77575E+06 4.78247E+06 4.77936E+06

The unit-cell dimensions and crack length are kept constant, but the strut thickness is varied. (c = 200lm, crack length = 0.03m anda = ±200 · 10�6).

Table 4Fracture toughness of graded and uniform foams

Crack length (m) Relative density at the crack-tip Decreasing density Increasing density Uniform density

0.01 0.06075 1.10144E+06 1.03627E+06 1.03485E+060.02 0.068608 1.25052E+06 1.18201E+06 1.18004E+060.03 0.076874 1.33362E+06 1.33619E+06 1.33465E+060.04 0.085536 1.49961E+06 1.49980E+06 1.49878E+060.05 0.094582 1.67268E+06 1.67266E+06 1.67220E+06

The unit-cell dimension is kept constant but the crack length is varied. (c = 200 lm, h0 = 40 lm and a = ±200 · 10�6)

0.0E+00

5.0E+05

1.0E+06

1.5E+06

2.0E+06

2.5E+06

3.0E+06

3.5E+06

0 0.05 0.1 0.15 0.2 0.25

Relative Density

Frac

ture

Tou

ghne

ss (

Pa-m

0.5 ) Decreasing Density

Increasing Density

Homogeneous

Fig. 13. Comparison of fracture toughness of graded and homogeneous foams having same density at the crack-tip. The unit-celldimension c is kept constant at 100 lm. The strut thickness h is varied from 9.66 lm to 28.4 lm.

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Table 5Comparison of the fracture toughness for varying unit-cell dimensions with constant strut thickness (h = 20 lm)

Set b c01 (m) c2 (m) c3 (m) Crack length in terms of number of

elementsRelative density at thecrack-tip

Fracture toughness (Pa-m1/2) % difference

Graded foam Homogeneous

10 0.0745806 9.62060E+05 9.61172E+05 0.09220 0.0776305 1.00630E+06 1.00531E+06 0.098

1 �0.15e�6 200e�6 100e�6 100e�6 30 0.0812704 1.05636E+06 1.05533E+06 0.09740 0.0856898 1.11501E+06 1.11279E+06 0.19950 0.0911693 1.18440E+06 1.18018E+06 0.35660 0.0981422 1.26090E+06 1.25952E+06 0.109

70 0.221965 2.05336E+06 2.04649E+06 0.33560 0.228608 2.15865E+06 2.16343E+06 0.22150 0.236846 2.30923E+06 2.29987E+06 0.405

2 0.15e�6 50e�6 50e�6 50e�6 40 0.247332 2.47076E+06 2.46204E+06 0.35330 0.261132 2.66760E+06 2.65985E+06 0.29120 0.280113 2.92063E+06 2.90924E+06 0.39010 0.307863 3.24507E+06 3.23901E+06 0.187

50 0.0912307 1.18404E+06 1.18051E+06 0.29840 0.0982215 1.26352E+06 1.26041E+06 0.246

3 0.15e�6 50e�6 100e�6 100e�6 30 0.107422 1.36134E+06 1.35823E+06 0.22820 0.120075 1.48715E+06 1.48204E+06 0.34410 0.138575 1.65340E+06 1.64829E+06 0.310

70 0.0450318 7.45601E+05 7.43076E+05 0.3394 0.15e�6 200e�6 150e�6 100e�6 50 0.0470359 7.89333E+05 7.85621E+05 0.470

30 0.0495308 8.39136E+05 8.35061E+05 0.48610 0.0527223 9.11742E+05 8.93837E+05 1.964

70 0.0218062 3.73104E+05 3.72085E+05 0.2735 0.15e�6 200e�6 200e�6 200e�6 50 0.0230945 3.94813E+05 3.92956E+05 0.470

30 0.0246984 4.19129E+05 4.17251E+05 0.44810 0.0267500 4.48072E+05 4.46171E+05 0.424

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pyloons of varying sizes in an epoxy matrix. They measured the fracture toughness of the foam using the coher-ent reflection gradient sensing technique and high-speed photography. Their results are shown in Fig. 15 bythe solid line.

Because the method of fabrication their material relative density ranges from 0.5 to 1.0. On the other handin cellular materials the density usually varies from 0.02 to 0.4 depending on the microstructure. Neverthelessthe trend seen in the present simulations (Fig. 13) agree qualitatively with available experimental results(Fig. 15). Hence the present work can be considered as the first step in simulating fracture in graded cellularmaterials in order to estimate their fracture properties.

Acknowledgements

This project is funded by the NASA CUIP Program (formerly URETI) Grant NCC3-994, and managed byNASA Glenn Research Center. Claudia Mayer is the program manager.

References

Anals, G., Santare, H.M., Lambros, J., 2000. Numerical calculation of stress intensity factors in functionally graded materials.International Journal of Fracture 104, 131–143.

Choi, S., Sankar, B.V., 2003. Fracture toughness of carbon foam. Journal of Composite Materials 37 (23), 2101–2116.Choi, S., Sankar, B.V., 2005. A micromechanical method to predict the fracture toughness of cellular materials. International Journal of

Solids & Structures 42 (5–6), 1797–1817.Delale, F., Erdorgan, F., 1983. The crack problem for nonhomogenous plane. Journal of Applied Mechanics 50, 609–614.

0.00E+00

5.00E+05

1.00E+06

1.50E+06

2.00E+06

2.50E+06

3.00E+06

3.50E+06

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Relative Density

Frac

ture

Tou

ghne

ss (

Pa-m

0.5 )

Set1 (FGF)

Set1 (Homogeneous)

Set2 (FGF)

Set2 (Homogeneous)

Set3 (FGF)

Set3 (Homogeneous)

Set4 (FGF)

Set4 (Homogeneous)

Set5 (FGF)

Set5 (Homogeneous)

Fig. 14. Comparison of fracture toughness of graded and homogeneous foams. The graded foams have varying unit-cell dimensions, butconstant strut cross-section h = 20 lm.

0.00

0.50

1.00

1.50

2.00

0 0.2 0.4 0.6 0.8 1

Relative Density

KIc (

MP

a-m

1/2 )

Fig. 15. Variation of fracture toughness of syntactic foams with relative density. The solid line represents the experimental results forrelatively higher densities (El-Hadek and Tippur, 2003). The dotted line indicates the projected trend for low-density graded foams.

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El-Hadek, M.A., Tippur, H.V., 2003. Dynamic fracture parameters and constraint effects in functionally graded syntactic epoxy foams.International Journal of Solids & Structures 40, 1885–1906.

Gibson, L.J., Ashby, M.F., 1998. Cellular solids: Structure and properties, 2nd ed. Cambridge University Press, Cambridge, UK, pp. 219–222.

Gu, P., Dao, M., Asaro, R.J., 1999. Simplified method for calculating the crack tip field of functionally graded materials using the domainintegral. Journal of Applied Mechanics 66, 101–108.

Madhusudhana, K.S., Kitey, R., Tippur, H.V., 2004. Dynamic fracture behaviour of model sandwich structures with functionally gradedcore. In: Proceedings of the 22nd Southeastern Conference in Theoretical and Applied Mechanics (SECTAM), Alabama.

Sih, G.C., Liebowbitz, H., 1968Mathematical Theories of Brittle Fracture, vol. 2. Academic Press, New York, pp. 67–190.

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